Optimization Methods in Finance

My Current Teaching @ National Tsing Hua University

Spring 2023, Instructor 

金融優化方法 Optimization Methods in Finance (QF527800)

Course Description & Audience: The course serves as a twin course of Robust and Stochastic Portfolio Optimization (QF527200). After a quick review of some key topics from linear algebra, the course will dive into the theory of optimization with a focus on its application in quantitative finance. Specifically, this course covers an extensive introduction to first-order optimization methods. The material coverage will be suitable for graduate students focusing on financial engineering, stochastic optimization, and quantitative trading. Many of the results will be presented in a definition-theorem-proof manner. Prerequisite mathematical maturity is essential for success in this course. The intended topics of the course are listed as follows.

• • Vector Spaces

  • Extended Real-Valued Functions

  • Subgradients

  • Conjugate Functions

  • Smoothness and Strong Convexity

  • The Proximal Operator

  • Spectral Functions

  • Strong Duality and Optimality Conditions

  • Numerical Optimization Algorithms

Prerequisites:  A student planning to take this course should have a fair familiarity with linear algebra and mathematical analysis. Familiarity with some numerical software such as Matlab or Python is also required. If in doubt about the background, the student should consult with the instructor.

Time and Place: Lectures are at T5T6T7, Room 732 TSMC Building. Meetings are also possible at other times by appointment.

Textbooks & References: Students will receive significant handout material to support the lectures. Below are some recommended textbooks; as we progress, other research papers will be assigned as reading tasks.

1. A. Beck, First-Order Methods in Optimization, SIAM, 2017.

2. J. Nocedal and S. J. Wright, Numerical Optimization, Springer, 2006.

3. A. Shapiro, D. Dentcheva, A. Ruszcynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, 2014.

4. G. Cornuejols, J. Pena, R. Tütüncü, Optimization Methods in Finance, Cambridge University Press, 2018.

5. D. G. Luenberger, Optimization By Vector Space Methods, Wiley, 1969.

Teaching Method: Lecture.

Homework: Approximately bi-weekly. 

Office Hours: By appointment.

Grading: The grade will be based on one midterm test (30% for each), homework (30%), and a special project (40%). The instructor may exercise discretion up to 10% in each grading category.

Course Material

• Week 01: 02/20

  • Introduction to Optimization

  • Vector Spaces I

• Week 02: 02/27

  • Vector Spaces II

• Week 03: 03/05

  • Extended Real-Valued Functions

• Week 04: 03/12

  • Extended Real-Valued Functions II: Convex Function

• Week 05: 03/19

  • Extended Real-Valued Functions III: Support Function

• Week 06: 03/26

  • Research Topic Due.

  • Subgradients I

• Week 07: 04/02

  • Subgradients II

• Week 08: 04/09

  • Subgradients III

• Week 09: 04/16

  • Subgradients IV

• Week 10: 04/23

  • Subgradients V: Optimality Conditions

• Week 11: 04/30

  • Subgradients VI: Optimality Conditions

  • Research Proposal Due

• Week 12: 05/07

  • Conjugate Functions

• Week 13: 05/14

  • Midterm/Final Exam

• Week 14: 05/21

  • Review of the Exam

• Week 15: 05/28

  • Conjugate Functions II

• Week 16: 06/04

  • Iterative Algorithm

  • Final Project Due

Handouts

• Chapter 1: Review of Vector Spaces

• Chapter 2: Extended Real-Valued Functions

• Chapter 3: Subgradients

• Chapter 4: Conjugate Functions

  • Additional Handout: Introduction to Iterative Algorithms

Supplementary Material (Handouts from QF527200): 

• Chapter 1: Introduction 

  • Additional Handout: A note on the limit of finite-dimensional l_p norm. 

• Chapter 2: Convex Sets

• Chapter 3: Convex Functions 

• Chapter 4: Convex Optimization 

• Chapter 5: Optimization Under Uncertainty: A Robust Approach 

• Chapter 6: Sensitivity of Mean-Variance Models and Black-Litterman Approaches 

• Chapter 7: Unconstrained Optimization Theory 

  • Additional Handout: A short note on duality theory

• Chapter 8: On Risk-Averse Optimization I: Semideviation, Value at Risk, and Coherent Risks 

• Chapter 9: On Risk-Averse Optimization II: Conditional Value at Risk 

• Chapter 10: On Multi-Period Portfolio Optimization I: Kelly Criterion 

Assignments

• Assignment 01

• Assignment 02

• Assignment 03

• Assignment 04 

• Assignment 05 

• Assignment 06

• Assignment 07

• Assignment 08  

• Assignment 09 (no need to submit)

Exams (and the Topics Covered)

Midterm/Final Exam: Everything up to Chapter 4.

Toolbox

CVXPY package for Python (or CVX for MATLAB) will be used frequently when solving some convex optimization problems.

• CVXPY website 

Overleaf: A user-friendly online LaTeX editor.

• Overleaf website

Writing Tips.

• Writing Tips for Ph.D. Students, J. H. Cochrane, University of Chicago.

Final Project Policies.

Your final project should include the following ingredients. 

• • Title

    • Ensure it is concise, informative, and reflective of the project's main findings.

  • Abstract

    • Provide a succinct summary of the problem, methodology, main results, and conclusions.

  •  Introduction 

    • Provide a literature survey explaining what has been done, then point out what is missing in the literature (For example, say there is some weakness of the existing approach, so you want to fix it by providing your approach. As another example, say there is an issue that no one has noticed, so you jump in. As the last example, you found a phenomenon from your experiment/simulation/dataset, so you formed your hypothesis and wish to test it. This should provide enough ground to motivate your work. Then, provide the difference between your approach and others', and compare and contrast. 

  • Problem Formulation or Preliminaries 

    • If your work involves a theoretical setting, provide it in this section. Define every notion and the corresponding notations in a consistent and precise way. You should tell the reader exactly the main problem you want to tackle and the methodologies you adopted.

  • Main results or Empirical studies 

    • The main claim or findings go first. Then, provide supporting evidence. For empirical studies, provide a detailed explanation about your data; e.g., where did you get the data? Why did you choose this dataset, not others? What do you see from the result? Are there any good or bad findings? If you run a simulation, describe how you conducted it precisely and concisely.

  • Concluding Remarks or/and Future Work

  • References

• Your final project should be no more than 10 pages of A4 paper.

• You are required to use LaTeX to write your project; Overleaf might be a handy editor.

Final Project Proposed by Students

1. 林昕宏, Optimal  Denoising for Financial Series: a Frequency Approach

2. 簡承威, Hierarchical Risk Parity with Wasserstein K-means Clustering for Portfolio Optimization

3. 黃得晉, Deep Hedging with Transaction Cost Model

4. 張履濃, Kelly Criterion in Different Market Regimes

5. 許藝懷, Wasserstein-Kelly Optimization Method (vs. Moment-Based Kelly Optimization Method): An Empirical Study

6. 楊則勛, Stochastic Programming Approaches for Portfolio Optimization

7. 任翊瑄, 以整數規劃替廣告設計排程

8. 許譽瀚, On Data-Driven Kelly Betting: A Taylor-Based Approximation Approach 

9. 甘容, Robust ELG Portfolio with Wasserstein Metric - Uncertainty Selection